Abstract

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n + 1)-dimensional manifolds into an n-dimensional manifold Nⁿ in terms of immersions with prescribed normal bundles. We compute that for Nⁿ = Rⁿ the cobordism group of simple fold maps is isomorphic to the direct sum of the (n− 1)-st stable homotopy group of spheres and the (n − 1)-st stable homotopy group of the space RP^∞. By using geometric invariants defined in the author’s earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group in terms of natural homomorphisms between cobordism groups of immersions. We also compute the ranks of the oriented bordism groups of simple fold maps.

Keywords

fold singularity, immersion, cobordism, simple fold map, stable homotopy group

Mathematical Subject Classification

Primary: 57R45

Secondary: 57R90, 57R42, 55Q45

Milestones

Received: 23 May 2008
Revised: 3 October 2008
Accepted: 20 October 2008
Published: 27 November 2008

Authors